Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ineccnvmo Structured version   Visualization version   GIF version

Theorem ineccnvmo 37739
Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)
Assertion
Ref Expression
ineccnvmo (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem ineccnvmo
StepHypRef Expression
1 relcnv 6097 . . 3 Rel 𝐹
2 id 22 . . . 4 (𝑦 = 𝑧𝑦 = 𝑧)
32inecmo 37737 . . 3 (Rel 𝐹 → (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥))
41, 3ax-mp 5 . 2 (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥)
5 brcnvg 5873 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝐹𝑥𝑥𝐹𝑦))
65el2v 3476 . . . 4 (𝑦𝐹𝑥𝑥𝐹𝑦)
76rmobii 3378 . . 3 (∃*𝑦𝐵 𝑦𝐹𝑥 ↔ ∃*𝑦𝐵 𝑥𝐹𝑦)
87albii 1813 . 2 (∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥 ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
94, 8bitri 275 1 (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 844  wal 1531   = wceq 1533  wral 3055  ∃*wrmo 3369  Vcvv 3468  cin 3942  c0 4317   class class class wbr 5141  ccnv 5668  Rel wrel 5674  [cec 8703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rmo 3370  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707
This theorem is referenced by:  ineccnvmo2  37742
  Copyright terms: Public domain W3C validator